Approximations to the initial segments of perturbed trajectories by a system of two Keplerian orbits

An intermediate orbit in which the initial position vector and its successive time derivatives to the third-order inclusive coincide with those of a real perturbed trajectory was studied. The intermediate motion is determined by the motion of a point along one Keplerian orbit (the epicycle) whose focus moves along another Keplerian orbit (the deferent) having a fixed focus. The intermediate orbit has third-order tangency to the real trajectory, ensures good approximation to the initial segments of this trajectory, and may be useful when studying the motion of minor bodies of the solar system.